# Circuits 2

### Main Reference Book

Tim Hanshaw, “Real Analog: An Introduction to Electrical Circuits”, free online access

Roland E. Thomas, Albert J. Rosa, and Gregory J. Toussaint
The Analysis and Design of Linear Circuits”, Wiley, 7th Edition, 2012

### Course Goals

• AC analysis of linear circuits to include circuit theorems via classical and transform techniques.
• Emphasis is placed on the Laplace transform, including use of pole-zero and Bode diagrams to analyze and design circuits, including multiple filters (single pole cascade, Butterworth, Chebychev), and step response circuits.
• Phasor applications to sinusoidal steady state analysis and AC power.
• Computer analysis software is used as an aid to circuit analysis.
• Laboratory program practicing time and frequency domain analysis and design techniques on step response and filter problems.
• Applications to instrumentation and circuits.

## Course Modules

• Qualitatively state the effect of energy storage on the type of mathematics governing a system
• Define transient response
• Write the mathematical expression for a unit step function
• Sketch the unit step function
• Sketch shifted and scaled versions of the unit step function
• Write the mathematical expression for a decaying exponential function
• Define the time constant of an exponential function
• Sketch a decaying exponential function, given the function’s initial value and time constant
• Use a unit step function to restrict an exponential function to times greater than zero
• Write the circuit symbol for a capacitor
• State the mechanism by which a capacitor stores energy
• State the voltage-current relationship for a capacitor in both differential and integral form
• State the response of a capacitor to constant voltages and instantaneous voltage changes
• Write the mathematical expression describing energy storage in a capacitor
• Determine the equivalent capacitance of series and parallel combinations of capacitors
• Sketch a circuit describing a non-ideal capacitor
• Write the circuit symbol for an inductor
• State the mechanism by which an inductor stores energy
• State the voltage-current relationship for an inductor in both differential and integral form
• State the response of an inductor to constant voltages and instantaneous current changes
• Write the mathematical expression describing energy storage in an inductor
• Determine the equivalent inductance of series and parallel combinations of inductors
• Sketch a circuit describing a non-ideal inductor
• Write the general form of the differential equation governing a first order system
• State, in physical terms, the significance of a differential equation’s homogeneous and particular solutions
• Define, from memory, the relationships between a system’s unforced response, zero-input response, natural response, and the homogeneous solution to the differential equation governing the system
• Define, from memory, the relationships between a system’s forced response, zero-state response, and the particular solution to the differential equation governing the system
• Determine the time constant of a first order system from the differential equation governing the system
• Write mathematical expressions from memory, giving the form of the natural and step responses of a first order system
• Sketch the natural response of a first order system from the differential equation governing the system and the system’s initial condition
• Sketch the step response of a first order system from the differential equation governing the system and the amplitude of the input step function
• Write the differential equation governing RC and RL circuits
• Determine the time constant of RC and RL circuits from their governing differential equations
• Determine the time constant of RC and RL circuits directly from the circuits themselves
• Determine initial conditions on arbitrary RC and RL circuits
• Write from memory the form of the natural responses of RC and RL circuits
• Determine the natural response of RC and RL circuits, given the governing differential equation and initial conditions
• Write the form of the differential equations governing forced first order electrical circuits
• Determine the time constant of a forced electrical circuit from the governing differential equation
• Write differential equations governing passive and active first order circuits
• Determine the differential equation governing the step response of a first order electrical circuit
• Write the form of the particular solution of a first order differential equation, to a step input
• Write the form of the step response of a first order electrical circuit
• Determine the final conditions (steady-state response) of a first order electrical circuit, to a step input
• Define DC gain for a circuit and relate it to the steady-state response to a step input
• Determine the step response of a first order electrical circuit from the governing differential equation, the initial conditions, and the final conditions
• Write differential equations governing second order circuits
• Define damping ratio and natural frequency from the coefficients of a second order differential equation
• Express the form of the natural response of an arbitrary second order system in terms of complex exponentials, the damping ratio, and the natural frequency
• Summarize the behavior of the complex exponentials in the system natural response for the damping ratio ranges below:
• Damping ratio greater than one
• Damping ratio less than one
• Damping ratio equal to one
• Write complex numbers in terms of complex exponentials
• Express sinusoidal signals in terms of complex exponentials
• Classify overdampedunderdamped, and critically damped systems according to their damping ratio
• Identify the expected shape of the natural response of over-, under-, and critically damped systems
• State from memory the definition of an underdamped second order system’s overshoot, rise time, and steady-state response
• Use the coefficients of a second order system’s governing equation to estimate the system’s overshoot, rise time, and steady-state response
• Video 1 – Introduction
• Video 2 – Flow Diagram
• Video 3 – s-Domain Models
• Video 4 – Algebraic Equation
• Video 5 – Initial and Final Value Theorem
• Video 6 – Partial Fraction Expansion 1
• Video 7 – Inverse Laplace
• Video 8 – Plot
• Video 9 – Partial Fraction Expansion 2
• Video 10 – Practice Example

### Videos

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• Define state variables for electrical circuits
• Write differential equations governing electrical circuits in state variable form
• Use MATLAB and/or Octave to simulate the impulse response of an electrical circuit
• Use MATLAB and/or Octave to simulate the step response of an electrical circuit
• Use MATLAB and/or Octave to plot the state trajectory of an electrical circuit
• State the relationship between the sinusoidal steady state system response and the forced response of a system
• For sinusoidal steady-state conditions, state the relationship between the frequencies of the input and output signals for a linear, time-invariant system
• State the two parameters used to characterize the sinusoidal steady-state response of a linear, time invariant system
• Express sinusoidal signals in phasor form
• Perform frequency-domain analyses of electrical circuits
• Sketch phasor diagrams of a circuit’s input and output
• State the definition of impedance and admittance
• State how to use the following analysis approaches in the frequency domain:
• KVL and KCL
• Voltage and current dividers
• Circuit reduction techniques
• Nodal and mesh analysis
• Superposition, especially when multiple frequencies are present
• Thévenin’s and Norton’s theorems
• Video 1 – Fourier Intro
• Video 2 – Fourier Series vs Fourier Transform
• Video 3 – Fourier Series – Alternative Form
• Video 4 – Fourier Coefficients
• Video 5 – Amplitude and Phase Spectrum
• Video 6 – Circuit Analysis
• Video 7 – Practice Circuit
• Video 8 – Fourier Transform Intro
• Video 9 – Fourier Transform (Path 1 part a)
• Video 10 – Fourier Transform (Path 1 part b)
• Video 11 – Fourier Transform (Path 2)
• Video 12 – Fourier vs Laplace
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• Use the frequency response of a system to determine the frequency domain response of a system to a given input
• State from memory the definition of signal spectrum
• Create plots of given signal spectra
• Plot a circuit’s magnitude and phase responses
• Check a circuit’s amplitude response at low and high frequencies against the expected physical behavior of the circuit
• Graphically represent a system’s frequency domain response from provided signal spectra plots and plots of the system’s frequency response
• Identify low pass and high pass filters
• Calculate a system’s cutoff frequency
• Determine the DC gain of an electrical circuit
• Write, from memory, the equation used to convert gains to decibel form
• Sketch straight-line amplitude approximations to Bode plots
• Sketch straight-line phase approximations to Bode plots
• Define instantaneous power, average power, and reactive power
• Define real power, reactive power, and complex power
• Define RMS signal values and calculate the RMS value of a given sinusoidal signal
• State, from memory, the definition of power factor and calculate the power factor from a given combination of voltage and current sinusoids
• Draw a power triangle
• Correct the power factor of an inductive load to a desired value